Let us start with the case when \({\mathcal {J}}_{z_0}(p)\ne \lbrace z_0\rbrace \) .
Assume that \(A_R(L)\)  is a spider's web for some sufficiently large \(R\) .
By definition,
there exists a sequence of bounded simply-connected domains \(G_n\)  such that
\(G_n\subset G_{n+1}\) , \(\partial G_n\subset A_R(L)\)  for \(n\in {\mathbb {N}}\) , and
\(\bigcup G_n={\mathbb {C}}\) . We can assume w.l.o.g. that every \(G_n\)  contains 0
(since this is true anyway for all sufficiently large \(n\) ).
By Proposition REF , for every \(n\in {\mathbb {N}}\) , the curve
\(L(\partial G_n)\)  intersects the filled Julia set of \(p\) .
Let \(K>0\)  be the radius of the smallest disk around 0
which contains the (filled) Julia set of \(p\) . Then there
exists a sequence of points \(w_n\in \partial G_n\)  such that
\(\vert L(w_n)\vert \le K\) . But this contradicts the assumption that
all points \(z\in \partial G_n\)  satisfy \(\vert L(z)\vert \ge \operatorname{M}(L,R)\) .
Let us now consider the situation when \({\mathcal {J}}_{z_0}(p)=\lbrace z_0\rbrace \) .
By [1]} it is sufficient to find
a sequence of bounded simply-connected domains \(G_n\)  such that
for all (sufficiently large) \(n\) ,
\(G_n\supset \lbrace  z\in {\mathbb {C}}: \vert z\vert <\operatorname{M}^n(L,R)\rbrace \) 

and
\(G_{n+1} \text{ is contained in a bounded component of }{\mathbb {C}}\setminus L(\partial G_n).\) 

Let \(R_1\)  be the constant from Lemma REF , and
set \(R:=\max \lbrace R_L,R_1\rbrace \) .
For \(n\in {\mathbb {N}}\)  let \(r_n:=\vert \lambda \vert ^n\operatorname{M}^n(L,R)\) 
(see also Lemma REF ).
By Lemma REF ,
there exists
a simple closed curve \(\Gamma ^{r_n}\)  separating \({\mathbb {S}}_{r_n}\)  and
\({\mathbb {S}}_{r_n^d}\)  such that \(\operatorname{m}(L, \Gamma ^{r_n})>\operatorname{M}(L,r_n)\) .
We define \(G_n\)  to be the interior of \(\Gamma ^{r_n}\) .
Then every \(G_n\)  is a bounded simply-connected domain with
\(G_n\supset \lbrace  z\in {\mathbb {C}}:\vert z\vert <r_n\rbrace \supset \lbrace  z\in {\mathbb {C}}:\vert z\vert <\operatorname{M}^n(L,R)\rbrace .\) 

Furthermore, it follows from Lemma REF  with \(m=d\)  that
\(\operatorname{m}(L,\partial G_n)=\operatorname{m}(L,\Gamma ^{r_n})>\operatorname{M}(L,r_n)>r_{n+1}^d>\max _{z\in \partial G_{n+1}}\vert z\vert ,\) 

hence \(G_{n+1}\)  is contained in a bounded component of \({\mathbb {C}}\setminus L(\partial G_n)\) 
and the claim follows.
